how much easier is it to estimate a mean than a minimum?

MAB.R

n_sim <- 10000

mu <- -10
sd <- 2
n  <- 30

z <- array(NA, c(3,n_sim))
for(i in 1:n_sim){
  x <- rnorm(n, mu, sd)
  z[1, i] <- mean(x)
  z[2, i] <- min(x)
  z[]
}
mean(z[1,])
mean(z[2,])

sd(z[1,])
sd(z[2,])

sd(z[1,])/mean(z[1,])
sd(z[2,])/mean(z[2,])

# "True" n-year min
qnorm(1/n, mu, sd)

# Median absolute bias (MAB)
median(abs(z[1,]-mu))
median(abs(z[2,] - qnorm(1/n, mu, sd)))

median(abs(z[2,] - qnorm(1/n, mu, sd))) / median(abs(z[1,] - mu))